随机$k$ NN图中Weakly Connected图的拷贝数分布
On the Distribution of the Number of Copies of Weakly Connected Digraphs in Random $k$NN Digraphs
摘要：
在有$n$顶点的有向图中，极小构造是有$m<<n$顶点的子有向图。研究了$\mathbb{R}^d$中随机点过程数据的$k$最近邻（$k$ NN）有向图中极小结构的拷贝数。基于齐次泊松过程和二项点过程下点集泛函的渐近理论，给出了极小构造数的渐近性质的一般结果，并作为推论，得到了$k$ NN有向图中固定阶数、共享$k$ NN对数和自反$k$ NN对数的渐近结果.
英文摘要：
In a digraph with $n$ vertices, a minuscule construct is a subdigraph with $m<<n$ vertices. We study the number of copies of a minuscule constructs in $k$ nearest neighbor ($k$NN) digraph of the data from a random point process in $\mathbb{R}^d$. Based on the asymptotic theory for functionals of point sets under homogeneous Poisson process and binomial point process, we provide a general result for the asymptotic behavior of the number of minuscule constructs and as corollaries, we obtain asymptotic results for the number of vertices with fixed indegree, the number of shared $k$NN pairs and the number of reflexive $k$NN's in a $k$NN digraph.
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